In the following table, the median age of 200 spectators of a football match is 32. Find the missing frequencies p and q. | Age (in years) | Number of Spectators | |---|---| | 0 - 10 | 20 | | 10 - 20 | p | | 20 - 30 | 50 | | 30 - 40 | 60 | | 40 - 50 | 32 | | 50 - 60 | q |

Step 1: Set up equations for total frequency and median class

First, we use the given total number of spectators to form an equation. The sum of all frequencies must equal 200. This gives us our first equation relating $p$ and $q$. Since the median age is 32, the median class is $30-40$.

Step 2: Calculate cumulative frequencies

To apply the median formula, we need to calculate the cumulative frequencies for each class interval. The cumulative frequency for a class is the sum of its frequency and the frequencies of all preceding classes.

Step 3: Apply the median formula

The median formula is used to find the median of grouped data. Here, $L$ is the lower limit of the median class (30), $N$ is the total frequency (200), $CF$ is the cumulative frequency of the class preceding the median class ($70+p$), $f$ is the frequency of the median class (60), and $h$ is the class size (10). We substitute the known values into the formula.

Step 4: Solve for p

Now, we simplify the equation from the previous step to solve for $p$. We perform algebraic manipulations to isolate $p$ and find its value.

Step 5: Solve for q

Finally, we substitute the value of $p$ (18) back into the first equation ($p + q = 38$) to find the value of $q$.

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